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Dispersion
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Dispersion occurs when pure plane waves of different wavelengths have different
propagation velocities. In the presence of dispersion, the wave velocity is no longer
uniquely defined, giving rise to the distinction of phase velocity and group velocity.
In a non-dispersive medium such as a vacuum, all frequencies of radiation, including
light, travel at the same speed regardless of wavelength. As a result a wavepacket will
not change shape as it travels.
In a dispersive medium different frequency components travel at different speeds.
As a result, a wavepacket spreads out because the longer wavelengths move faster and
the shorter wavelengths lag behind. So even if we start with a fairly localized particle,
it will soon loose this localization.
In the study of solids, the study of the dispersion relation of electrons is of paramount
importance. The periodicity of crystals means that many levels of energy are possible for
a given momentum and that some energies might not be available at any momentum. The
collection of all possible energies and momenta is known as the band structure of a material.
Properties of the band structure define whether the material is an insulator, semiconductor
or conductor.
Phonons are to sound waves in a solid what photons are to light: They are the quanta that
carry it. The dispersion relation of phonons is also important and non-trivial. Most
systems will show two separate bands on which phonons live. Phonons on the band that
cross the origin are known as acoustic phonons, the others as optical phonons.
The speed of a plane wave, v, is a function of the wave's wavelength:
v = λf
ω = 2πf
k = 2π/λ
=> ω = vk
Monochromatic light in vacuum: ω = kc
Monochromatic light in medium: ω = kc/n where n = refractive index.
Free, non-relativistic quantum mechanical particle of mass m: E_{k} = hω_{k} = h^{2}k^{2}/2m
=> ω_{k} = hk^{2}/2m